Factor the quadratic expression completely. $8x^2-18x-5=$
Explanation: Since the terms in the expression do not share a common monomial factor and the coefficient on the leading $x^2$ term is not $1$, let's factor by grouping. The expression ${8}x^2{-18}x{-5}$ is in the form ${A}x^2+{B}x+{C}$. First, we need to find two integers ${a}$ and ${b}$ such that: $\begin{cases} &{a}+{b}={B}={-18} \\\\ &{ab}={A}{C}= ({8})({-5})=-40 \end{cases}$ We find that ${a}={2}$ and ${b}={-20}$ satisfy these conditions, since ${2}+({-20})={-18}$ and $({2})({-20})=-40$. Next, we can use these values to rewrite the $x$ -term and factor by grouping. $\begin{aligned} 8x^2-18x-5&=8x^2+{2}x{-20}x-5 \\\\ &=2x(4x+1)-5(4x+1) \\\\ &=(4x+1)(2x-5) \end{aligned}$ In conclusion, $8x^2-18x-5=(4x+1)(2x-5)$